An ESP32-C3 Super Mini can be found for below 3$ (cheapest I had was 1.58€). Since the original clock is 3.88$, it can't be that much cheaper.

You often have a radio clock source like DCF77 that all those radio controlled clocks use

ESPs are so cheap that you couldn't possibly save very much money, and the way economies of scale work it may or may not be cheaper to use NFC anyways.

We've been shopping for a simple bathroom clock to replace our final Amazon Echo and leave that increasingly dystopian ecosystem. There are some models that use Bluetooth on your phone to sync the time. I could imagine BLE being a good low-power and relatively stateless solution. But given our goals, we're not going to install an app on a phone just to maintain a wall clock. (I'd be fine if Android provided BLE time sync as a built-in service.)